Are "Conditional apology" and "poisoned apology", rude?

I've heard

I'm sorry your frog is dead.
I'm sorry if your frog's death causes you pain.
I'm sorry my taunting you about your frog's death caused you pain. You should seek therapy.

Do the second and third type of apology ("Conditional apology" and "poisoned apology") have something with it? I mean, does the speaker really feel sorry?

Are these kind of apologies, more rude than polite?

EDIT:

I asked the question cause of this article:

An apology should give the sense that you actually feel some form of regret. "Sorry if" is a conditional apology. Conditional apologies make things worse, not better.

I find your examples confusing, but looking at the article and your question here is what I have to say:

Conditional apologies indeed leave space for possibility that there is nothing to apologize for.

This in itself is not rude.

Actually, I believe that such constructs became popular because it allows you to be extra polite and to apologize for minor things which might or might not have caused any discomfort to someone.

In this light, conditional apologies are best used not as actual apologies, but as a polite question, e.g:

"I am sorry if the spices are not to your liking."

to which a reply might be

"Don't be silly, it is delicious."

or

"It is fine, I am just not so used to them."

to which then actual apology is given

"I am so sorry I completely forgot that you are actually not from these parts. Please have some fish, it is not hot at all."

Both branches of conversation are polite and respectful.

So, while it is not an actual apology, but only implies it conditionally, it is definitively not rude on its own.

Context determines it - it will be seen as rude if:

• direct apology is obviously necessary ("I am sorry if my running over your dog caused you any pain.", "I am sorry if my insults to your mother bother you." - here what is insulting is not the conditional, but the implication that you actually might not care)
• is used sarcastically (both of previous examples might be sarcastic, with intent to hurt, but notice that context can be imagined in which intention of hurt is not present and the choice is simply due to clumsiness or actual state of facts - if for example the person addressed wanted to get rid of the dog or if he is not even talking to their mother)

A "poisoned" apology is a statement that looks like an attempt to apologize on the surface, but is in reality just a further jab at the other person.

The first attempt is a genuine apology; the speaker is sorry that the listener's frog is dead.

The second attempt is marginal; the speaker is not sorry that the frog is dead, but is sorry about any pain that it may cause. If the speaker killed the frog, this could be construed as a poisoned apology; the speaker is not sorry about his actions, which cause the speaker pain, but only about the pain. The speaker would be unlikely to change his behavior should the situation repeat.

The last apology is nothing of the sort; the speaker is clearly either sarcastic or sociopathic, in that they first obviously taunted the listener about a death, then suggested they look elsewhere for help.

Other poisoned apology examples might include the very-common one, "I'm sorry you feel that way". Perfectly polite, probably a true statement, but not an apology by any stretch of the term. -